Is it possible to express $\; "\!n\pmod 3\!"\;$ with combinations of the functions plus, minus, multiplication, division and exponentiation in $\mathbb C$ or preferably in $\mathbb Z[i]$?
I'ts not allowed with modulo or integer parts, and the inverses are only valid if they can be expressed as above. Also all sums, products etc must be finite.
There is a simular question Closed-Form Modular Arithmetic where trigonometric functions and logarithms are allowed.
If it isn't possible, how to prove that?
If complex exponentiation is OK, then you may do it. The map $f: n \mapsto \exp(2\pi n/3)$ takes ${\mathbb Z}$ to the three cubic roots of $1$: $1$, $\gamma=\exp(2\pi i /3)$ and $\overline{\gamma}$. Now pick a polynomial $P$ that maps $1$ to $0$, $\gamma$ to $1$ and $\overline{\gamma}$ to $2$. Then $P\circ f$ will do the job. Same principle for mod $N$.
Later edit: In the general case set $\gamma=\exp(2\pi i/N)$ and define $e_N(z)=\frac{1}{N} \left(z^{N-1}+z^{N-2} + \cdots +z +1\right) $. Then $e_N$ maps $1$ to $1$ but every $\gamma^k$, $1\leq k<N$ to zero. The wanted function is then: $$ h_N(n) = \sum_{k=1}^N k \ e_N \left( \ \gamma^{n-k} \right), $$ which maps $n$ to $n\ (\mbox{mod}\; N)$. What is really happening here is that we are doing a forward + backward Fourier transform of the modulus function.